no. 1
On the joint spectral radius for isometries of non-positively curved spaces and uniform growth
[Sur le rayon spectral joint pour les groupes d’isométries des espaces à courbure négative ou nulle et la croissance uniforme]
Breuillard, Emmanuel1 ; Fujiwara, Koji2
1 University of Cambridge, DPMMS Wilberforce Road CB3 0WB Cambridge (U.K.)
2 Department of Mathematics, Kyoto University Kitashirakawa Oiwake-cho, Sakyo-ku, Kyoto 606-8502 (Japan)
Annales de l'Institut Fourier, Tome 71 (2021) no. 1, pp. 317-391.

Nous généralisons la notion de rayon spectral joint dans le cadre des actions de groupes par isométries sur les espaces à courbure négative ou nulle et nous donnons des versions géométriques des résultats de Berger–Wang et de Bochi valables dans tout espace δ-hyperbolique ainsi que dans les espaces symétriques de type non-compact. Cette méthode permet de produire des éléments hyperboliques dans de nombreuses situations géométriques classiques. Nous donnons par ailleurs des applications à la croissance uniforme ainsi qu’une nouvelle preuve et une généralisation d’un théorème de Besson–Courtois–Gallot.

We recast the notion of joint spectral radius in the setting of groups acting by isometries on non-positively curved spaces and give geometric versions of results of Berger–Wang and Bochi valid for δ-hyperbolic spaces and for symmetric spaces of non-compact type. This method produces nice hyperbolic elements in many classical geometric settings. Applications to uniform growth are given, in particular a new proof and a generalization of a theorem of Besson–Courtois–Gallot.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3374
Classification : 20F65, 20M20, 15A30
Keywords: joint spectral radius, $\mathrm{CAT}(0)$ spaces, Gromov hyperbolic spaces, uniform exponential growth, affine isometric actions, ping-pong
Mot clés : rayon spectral joint, espaces $\mathrm{CAT}(0)$, espaces hyperboliques au sens de Gromov, croissance exponentielle uniforme, actions isométriques affines, ping-pong

Breuillard, Emmanuel 1 ; Fujiwara, Koji 2

1 University of Cambridge, DPMMS Wilberforce Road CB3 0WB Cambridge (U.K.)
2 Department of Mathematics, Kyoto University Kitashirakawa Oiwake-cho, Sakyo-ku, Kyoto 606-8502 (Japan)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     title = {On the joint spectral radius for isometries of non-positively curved spaces and uniform growth},
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Breuillard, Emmanuel; Fujiwara, Koji. On the joint spectral radius for isometries of non-positively curved spaces and uniform growth. Annales de l'Institut Fourier, Tome 71 (2021) no. 1, pp. 317-391. doi : 10.5802/aif.3374. https://aif.centre-mersenne.org/articles/10.5802/aif.3374/

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